luca190501-operational_tangent_space.tex

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%% Author: PGL  Porta Mana
%% Created: 2015-05-01T20:53:34+0200
%% Last-Updated: 2019-05-02T00:57:23+0200
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Paper's details
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{\propertitle}
\author{%
\hspace*{\stretch{1}}%
%% uncomment if additional authors present
% \parbox{0.5\linewidth}%\makebox[0pt][c]%
% {\protect\centering ***\\%
% \footnotesize\epost{\email{***}{***}}}%
% \hspace*{\stretch{1}}%
\parbox{0.5\linewidth}%\makebox[0pt][c]%
{\protect\centering P.G.L.  Porta Mana\\%
\footnotesize\epost{\email{piero.mana}{ntnu.no}}}%
\hspace*{\stretch{1}}%
%\quad\href{https://orcid.org/0000-0002-6070-0784}{\protect\includegraphics[scale=0.16]{orcid_32x32.png}\textsc{orcid}:0000-0002-6070-0784}%
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\date{Draft of \today\ (first drafted \firstdraft)}
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\begin{abstract}\labelsep 0pt%
  \noindent ***
\\\noindent\emph{\footnotesize Note: Dear Reader
    \amp\ Peer, this manuscript is being peer-reviewed by you. Thank you.}
% \par%\\[\jot]
% \noindent
% {\footnotesize PACS: ***}\qquad%
% {\footnotesize MSC: ***}%
%\qquad{\footnotesize Keywords: ***}
\end{abstract}
\selectlanguage{british}\frenchspacing

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\section{***}
\label{sec:***}

Imagine to stand on a differential manifold and to be armed with a pen and
with some kind of instrument that can tell you the value of every function
and of the parameter of every curve at each point of the manifold.

For a being thus equipped, what is a tangent vector? how is the sum of two
tangent vectors performed? and what about other tangent objects and other
operations with them?

We know that a differential manifold locally looks like $\RR^{n}$, by
definition. We know the various definitions of tangent vector and tangent
space. The latter is sometimes represented as a portion of $\RR^{n}$ -- for
example, a portion of a plane -- tangent to a point of the manifold. But
when we \enquote{stand} on that point there's no such plane to be seen.
Moreover, how can we conceive of straight lines, even very short ones, if
we only have a notion of smoothness and no notions of parallelism or
distance?

In this note I present, very informally, a practical way to think about
tangent vectors, tangent covectors, and their operations; and about some
other tangent objects and their operations. It may help students of
differential geometry get an intuitive and visual understanding of tangent
objects. It's a way of thinking inspired by the illuminating graphical
representation discussed by Burke \citep[\sect~II.8, especially
pp.~62--65]{burke1985_r1987}. Unlike Burke I won't introduce coordinate
charts. In fact, the notion of chart will naturally appear, as a derived
object, from the way of thinking presented here; this derivation bears some
similarities with Chevalley's \citey[\chap~III]{chevalley1946} and Penrose
\amp\ Rindler's \citey[\sect~4.1]{penroseetal1984_r2003} definition of
differential manifold via a ring of functions (compare the first footnote
on p.~181 of the last reference).



% Tangent vectors and other tangent objects are the building blocks of
% other notions. For example, if we live on a manifold with a metric $\yg$
% and want to know the length of a curve between two points on the manifold,
% we have to integrate the expression $\sqrt{\yg(\yv,\yv)}$, where $\yv$ is
% a  vector tangent to the curve at each point.
% But how would we \emph{see}
% this tangent vector? and the metric tensor? And how would we operationally
% construct the sum of two tangent vectors or apply the metric tensor to two
% of them?


\textcolor{white}{If you find this you can claim a postcard from me.}

*** in Rn or Euclid space have definite image. In tg plane depiction, how
long shall we draw them?

%%\setlength{\intextsep}{0.5ex}% with wrapfigure
%\begin{figure}[p!]%{r}{0.4\linewidth} % with wrapfigure
%  \centering\includegraphics[trim={12ex 0 18ex 0},clip,width=\linewidth]{maxent_saddle.png}\\
%\caption{caption}\label{fig:comparison_a5}
%\end{figure}% exp_family_maxent.nb


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%%% Acknowledgements
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\iffalse
\begin{acknowledgements}
  \ldots to Mari \amp\ Miri for continuous encouragement and affection, and
  to Buster Keaton and Saitama for filling life with awe and inspiration.
  To the developers and maintainers of \LaTeX, Emacs, AUC\TeX, Open Science
  Framework, Python, Inkscape, Sci-Hub for making a free and unfiltered
  scientific exchange possible.
%\rotatebox{15}{P}\rotatebox{5}{I}\rotatebox{-10}{P}\rotatebox{10}{\reflectbox{P}}\rotatebox{-5}{O}.
\sourceatright{\autanet}
\end{acknowledgements}
\fi

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%     \arxivp \mparcp \philscip \biorxivp}}

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]

\end{document}

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